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Gravity equations:Workhorse, toolkit, cookbook
Keith Head1,3 Thierry Mayer2,3,4
1Univ. of British Columbia, Sauder School of Business
2Sciences Po
3CEPR
4CEPII
National Bank of Belgium, November 21st, 2012.
Why you should not resist gravity
Gravity equations are a model of bilateral interactions in which sizeand distance effects enter multiplicatively.
They have been used as a workhorse for analyzing the determinantsof bilateral trade flows for 50 years.
Gravity is a tool for evaluating welfare impacts of trade policychanges.
Gravity also works for FDI, portfolio investment, tradable services,migration, tourism, even the internet.
“Physics envy”: Gravity is rare example in economics of “law-like”behavior.
Gravity research exemplifies the beneficent roles of empiricalregularities in guiding theory development and theory in guidingestimation.
Japan’s EU trade is GDP-proportionate
MLT
ESTCYP
LVA
LTUSVN
SVK
HUNCZE
PRT
FINIRLGRC
DNK
AUTPOL
SWE
BELNLD
ESP ITAFRA
GBRDEU
slope = 1.001fit = .85
.05
.1.5
15
10Ja
pan'
s 20
06 e
xpor
ts (G
RC =
1)
.05 .1 .5 1 5 10GDP (GRC = 1)
MLT
EST
CYP
LVA
LTU
SVN
SVK
HUNCZE
PRT
FIN
IRL
GRC
DNKAUT
POL
SWEBELNLD ESP
ITAFRAGBR
DEU
slope = 1.03fit = .75
.51
510
5010
0Ja
pan'
s 20
06 im
ports
(GRC
= 1
).05 .1 .5 1 5 10
GDP (GRC = 1)
(a) exports, 2006 (b) imports, 2006
France’s trade-distance relationship
slope = -.683fit = .22
.005
.05
.1.5
15
10Ex
ports
/Par
tner
's G
DP
(%, l
og s
cale
)
500 1000 2000 5000 10000 20000Distance in kms
EU25EuroColonyFrancophoneother
slope = -.894fit = .2
.005
.05
.1.5
15
1025
Impo
rts/P
artn
er's
GD
P (%
, log
sca
le)
500 1000 2000 5000 10000 20000Distance in kms
EU25EuroColonyFrancophoneother
(c) Exports, 2006 (d) Imports, 2006
... and yet
“If I had access to captive research assistance and funds, Icould examine whether, for all conceivable combinations ofcountries and distances among them, and for several differenttime periods, the premise [that proximity increases trade] isvalid. I do not, so I must rely on casual empiricism and a prioriarguments...Borders [such as the one between Pakistan andIndia] can breed hostility and undermine trade, just as alliancesbetween distant countries with shared causes can promotetrade... ...[The premise that distance reduces trade] does nothave a firm empirical or conceptual basis.” Bhagwati (1993)
India & Pakistan’s deviations from gravity
PAK-->IND
4
4
44
4
44
4
44
44
4
4
4
4
44
4
4
44 4
444
4
4
444
5
5
5
PAK-->GBR
.00
01
.00
1.0
1.0
5.1
.51
5
1950 1960 1970 1980 1990 2000
real/naive gravity-pred. trade
IND-->PAK
44
4
44
44
444
4
4 44
4
4
4
444
44444
4
4444
4
5
5
5
IND-->GBR
.00
1.0
1.0
5.1
.51
5
1950 1960 1970 1980 1990 2000
real/naive gravity-pred. trade
(a) Pakistan (b) India
Outline of rest of talk
1. The ascent of gravity: landmark years
2. Defining gravity: harder than you’d think
3. Microfoundations: an “embarrassment of riches”
4. Theory-consistent estimation (1st Monte Carlo)
5. Trade impacts: meta-analysis, GETI, & welfare
6. Gravity’s errors: the heteroskedasticity challenge (2nd Monte Carlo)
7. Causes and consequences of zeros (3rd Monte Carlo)
8. Extensive & intensive margins of gravity
9. Future directions for gravity research.
State of affairs in 1995:
Leamer and Levinshon (1995, HIE): gravity models “have producedsome of the clearest and most robust findings in economics. Butparadoxically they have had no effect on the subject ofinternational economics.”
... “Why don’t trade economists ‘admit’ the effect of distance intotheir thinking?”
One might add: “... still, 40 years after Isard and Peck?”
3 landmarks steps to recognition
Admission (1995): Gravity is one way to measure the largeamount of “missing trade” and explain it. LL (1995). Trefler(1995), McCallum (1995).
The MR/fixed effects revolution (2002-2004): Gravity has(many) micro-foundations + easy to do “structural” estimation. EK(2002), AvW (2003), Feenstra (2004), Redding and Venables(2004).
Convergence with the het. firms lit. (2007-2008): Gravitycompatible with new paradigm, new usage of the tool to measurethe margins. BJRS (2007), HMR (2008), Chaney (2008), MO(2008).
Defining gravity
3 definitions:
1. General structural gravity
2. Special structural gravity
3. Naive gravity
General structural gravity
Set of models that yield bilateral trade equations that can beexpressed as
Xni = GSiMnφni .
Si : “capabilities” of exporter i
Mn : characteristics of a country which make it a largeimporter.
0 ≤ φni ≤ 1 : bilateral accessability of destination market n toexporter i (combines trade costs with their respectiveelasticity).
Special structural gravity
Subset of general structural gravity models in which bilateral tradeis given by
Xni =Yi
ΩiSi
Xn
ΦnMn
φni ,
where Yi =
n Xni is the value of production, Xi =
i Xni is thevalue of expenditure, and Ωi and Φn are “multilateral resistance”terms defined as
Φn =
φnY
Ωand Ωi =
φiX
Φ.
Requirements of Special Structural Gravity
1. Budget shares (πni = Xni/Xn) multiplicatively separable:
πni =Siφni
Φn, where Φn =
Sφn.
2. Market clearing (production = sum of shipments to alldestinations):
Yi =
n
Xni = Si
n
φniXn
Φn= SiΩi .
Naive gravity
Naive gravity equations express bilateral trade as
Xni = GY ai Y
bn φni
Imposes the implausible restriction that φni is a constant.
Baldwin and Taglioni (2007): omission of 1/(ΩiΦn) is the“gold medal mistake” of gravity equations, characterizingmost empirical work before Anderson and van Wincoop(2003).
Special gravity’s many micro-foundations
Si Mn φni Exp. Imp. Bilat. Tr. elas.
CES NPD A−i w
i Xn/Φn τ ni 1− σCES MC DSK Niw
i Xn/Φn τ ni 1− σ
CES MC CET Liwγ1+γ
i Xn/Φn τ ni(1+γ)(1−σ)
σ+γ
Het. consumers A−i Niw
i Xn/Φn τ nia−ni −θ
Het. inds. EK Tiwβi Φ1−β
i Xn/Φn τ ni −θ
Het. firms CES Ni αiw
−µ[ θσ−1
−1]i Xn/Φn τ niξ
θσ−1
−1
ni −θHet. firms Ni α
iwi Xn/Φn τ ni −θ
“log-concave”
Not special: Examples of general gravity
Si Mn φni Exp. Imp. Bilat. Tr. elas.
CES NPD A−i w
i 1 τ ni 1− σ(outside good)Het. firms Ni α
iwi Lnc∗θ+2
n τ ni −θ(linear w/ OG)Het. firms Liµmax
i w i Ln(wnτnnmd
n )θ+1 τ ni −θ
(CARA)
Gravity for other flows
Human movement: migration, commuting, and tourism.
Anderson (2011, Ann. Rev.) presents a migration gravity modeldrawing on discrete choice techniques. Ahlfeldt et al. (2010) drawon Eaton and Kortum (2002) to specify a commuting gravity model.
Service offshoring. Head et al. (2009) adapt the Eaton andKortum (2002) model to the case of service offshoring.
Cross-border asset ownership. Martin and Rey (2004) propose a2-country model that they use to justify a gravity equation forbilateral portfolio investment. Head and Ries (2008) consider amodel generating a gravity equation for FDI, when it takes the formof acquisitions.
Remoteness
A few studies have included proxies for 1/Ωi and 1/Φn andreferred to them as “remoteness.”
Helliwell (1998) measures remoteness asREM1n =
i Distni/Yi . However, as Yi → 0, REM1
explodes.
A better measure of remoteness isREM2n = (
i Yi/Distni )−1.
Supposing φni ∼ Dist−1
ni and Xn = Yn, the correct Φn and Ωi
are
(Y/Distn)Ω−1
,
(Y/Distn)Φ−1
.
Still far off the mark.
Iterative structural estimation
1. Assume initial values of Ωi = 1 and Φn = 1,
2. Estimate the vector of parameters determining φni ,
3. Use a “contraction mapping” algorithm to find fixed pointsfor Ωi and Φn given those parameters.
4. Run OLS using lnXni − lnYi − lnXn + ln Ωi + ln Φni as thedependent variable. This gives a new set of φni parameterestimates.
5. Iterate until the parameter estimates stop changing.
Fixed effects estimation
Standard estimating procedure involves taking logs of the general gravityequation:
lnXni = lnG + ln Si + lnMn + lnφni .
Tradition: using log GDPs (and possibly other variables) as proxiesfor the ln Si and lnMj : B&T’s “Gold medal” mistake.
Since Harrigan (1996) practice has been moving towards using fixedeffects for these terms instead.
Note that it does not involve strong structural assumptions on theunderlying model. Only need general structural gravity to estimateφij consistently
Furthermore, market-clearing does not affect the estimationprocedure.
Can help control for country-specific patterns (entrepot trade...)
Monte Carlo study of different estimators
Monte Carlo using special structural gravity as a DGP.
We use actual data for 170 countries that have GDP,distance, and RTA data in 2006.
φni = exp(− ln Distni + 0.5RTAni )ηni .
2 types of missing values: suppress X% of observationsrandomly, or smallest X% of the initial set of export flows.
Estimators considered in 1st Monte Carlo
Abbrev. Description Introduced byOLS Linear-in-logs with GDPs Tinbergen (1962)SILS Structurally Iterated Least Sq. Anderson & van Wincoop (2003)∗
2WFE 2-way country fixed effects Harrigan (1996)DDM Double-Demeaning noneBVU Bonus Vetus OLS, simple av. Baier & Bergstrand (2010)BVW Bonus Vetus OLS, GDP-wts Baier & Bergstrand (2009)Tetrads Ratios of reference Head, Mayer & Ries (2010)
exporter & importer
Monte Carlo results
1. OLS badly biased under structural gravity DGP.
2. SILS is close from 2WFE (slightly less precise) and robust tomissings: not worth the computational effort?
3. DDM is one of the worse estimators when there are large numbersof non-random missing observations. BVU (related) appears to havebetter robustness properties.
4. BVW: not robust to missing data and very imprecise (high standarddeviation of the coefficients).
5. Tetrads: unbiased except when high numbers of randomly missingobservations.
6. Methods around computational problems (BV, Tetrads): mainlyobsolete due to new iterative methods for 2WFE?
Meta-analysis of gravity coefficients
Use Disdier and Head (2008) as a base
+ add other covariates
+ update since 2006–2012 (top5 + JIE + Restat)
+ add all price elasticity gravity papers found.
The final dataset includes a total of 161 papers, and more than2500 usable estimates.
Meta-analysis of gravity coefficients
All Gravity Structural GravityEstimates: median mean s.d. # median mean s.d. #
Origin GDP .97 .98 .42 700 .86 .74 .45 31Destination GDP .85 .84 .28 671 .67 .58 .41 29Distance -.89 -.93 .4 1835 -1.14 -1.1 .41 328Contiguity .49 .53 .57 1066 .52 .66 .65 266Same language .49 .54 .44 680 .33 .39 .29 205Colonial link .91 .92 .61 147 .84 .75 .49 60RTA/FTA .47 .59 .5 257 .28 .36 .42 108Same currency .87 .79 .48 104 .98 .86 .39 37Home 1.93 1.96 1.28 279 1.55 1.9 1.68 71“Structural gravity” here ≡ use of country fixed effects or ratio-type method.
Comments on meta-results
1. Destination GDP elasticities look slightly low.
2. RTA effects in line with existing meta
3. CU on the high side. See SST An. Rev. for estimate of ≈ 0.
4. Is there an inverse proportionality law (Xni ∼ 1/Dni ) for distance?(for a model of why there might be, see Chaney)
5. naive vs structural estimates of distance in line with predictionsfrom Monte, but not RTA
6. important endogeneity issues which call for dyadic FE inclusion.(not investigated here, see Baier & Bergstrand)
Price-shifter elasticities
“Gravity-based” estimates: regressing bil. trade on measures ofbilateral trade costs or exporter “competitiveness” (wages orproductivity). Typical equation:
lnXni = ln Si + lnMn + ln τni .
Recall that τni = pni/pfobi so one way to measure ln τni is the log ofone plus the ad valorem tariff rate charged by n on goods from i .
Estimates of the trade cost elasticity ()
Estimates: median mean s.d. #
Full sample: -4.76 -6.23 8.66 508
Estimation method:Naive gravity -2.66 -2.88 1.64 32Structural gravity
Country FEs -4.4 -5.19 6.59 347Ratios -7.07 -9.88 12.65 129
Identifying variable:Tariffs/Freight rates -5.51 -7.5 9.6 369Price/Wage/Exchange rate -1.23 -2.88 3.81 139
Notes: The number of statistically significant estimates is 508, obtained from 19 papers.
Tariff-equivalence of policy dummies.
lnXni = ln Si + lnMn + ρRTAni + ln τni .
Equivalence: ρ = (ln τMFNni − ln τRTAni )
Denote t MFN tariffs removed by RTA, and ν the ad-valoremtariff-equivalent of remaining trade barriers: τMFN
ni = 1 + ν + t andτRTAni = 1 + ν.
⇒ t = (1 + ν)[exp(ρ/)− 1].
Meta: ρ = 0.47 and tariff-based = 5.51, assuming ν = 0 impliest = 8.9%.
“Home” median coefficient is 1.93 ⇒ν = exp(1.93/5.51)− 1 = 42% ⇒ t = 12.6%.
WDI: 3.83% = weighted world MFN tariff in 2011. In 2000, theworld simple average MFN tariff is 12.8%.
3 measures of trade impact
Suppose lnφni is linear in Bni with coefficient β. What is the impact ontrade of changing Bni to B
ni?
Partial Trade Impact:
PTIni = φni/φni = exp[β(B
ni − Bni )].
Modular Trade Impact:
MTIni =X ni
Xni= exp[β(B
ni − Bni )] PTI
× Ωi
Ωi
Φn
Φn
MR adj.
General Equilibrium Trade Impact:
GETIni =X ni
Xni= exp[β(B
ni − Bni )] PTI
× ΩiΦn
ΩiΦ
n
MR adj.
× Y i X
n
YiXn GDP adj.
=Yi Xn
Ωi Φn
φni
GETI: GDP adj. algorithm
Production: Yi = wiLi , with Li constant, wi = Yi .
Xn = Yn, because of trade deficits, denoted as Dn. Assume that deficit isexogenously given on a per capita basis, that is Dn = Lndn. With thisassumption, Xn = wnLn(1 + dn), so that Xn = wn = Yn.
When πni obeys special structural gravity,
πni =(Yi τni )
πn(Yτn).
Using the market clearing condition that Y i =
n π
niX
n, one can solve
for the changes in production of each origin country.
Yi =1
Yi
n
πniπni YnXn =1
Yi
n
πni Y i φni
πnY φn
YnXn,
GETI/welfare 4-step program
1. Estimate a gravity equation, with dummy Bni indicating RTA/CU withcoefficient β and the trade elasticity, , or use results from ourmeta-analysis.
2. Calculate PTIni = φni = exp(β) for the ni for whom Bni = 1 andφni = 1 for all other pairs.
3. Plug estimated φni (along with initial values of Yi , Xn, and the πni )into Yi eqn. Substitute the φni and Y
i into πni eqn =⇒ matrix of tradechanges. Iterate using a dampening factor until πni stops changing.
4. Calculate GETI for each country pair = πni Yn and the welfare change= π1/
nn . (ACR)
PTI, MTI, GETI and welfare effects
Use TradeProd data for 2000 (square mfg. trade and productiondata for 84 countries).
Use = −4.76 in GETI and welfare.
median values of the real/counterfactual (links turned off) traderatio for relevant set of countries.
coeff PTI MTI GETI Welfaremembers: yes no yes no yes no
RTA/FTA .36 1.43 1.19 .94 1.28 .95 1.02 1.00Same currency .86 2.36 1.69 1.02 1.99 1.00 1.02 1.00Same language .39 1.48 1.35 .97 1.37 .99 1.01 1.00Colonial link .75 2.12 2.00 .97 2.06 .99 1.00 1.00Home (border) 1.90 6.69 6.18 .90 4.30 .66 .73 .
Comments on trade impacts
1. MTI (member) < PTI across the board =⇒ changes in MRstend to dampen trade impacts.
2. in general GETIs are very similar to MTI.
3. an advantage of both MTI and GETI is finding non-memberimpacts.
4. Strong trade impacts may have weak welfare effects. Why?initial πni so small for CU and colonies.
5. 27% welfare losses from border effects. Distortion towardsinternal flows is a big deal (we think) because πnn high due tolow distance to self.
3 topics w/ “unsettled” questions
1. Gravity’s errors
2. Causes and consequences of zeros
3. Firm-level gravity, extensive and intensive margins
Gravity’s errors
SST (2006) argue that Poisson PML is an attractive alternative tolinear-in-logs OLS if the variance of the error term depends uponthe RHS.
Poisson PML: Mean-Variance proportionality.
Gamma PML: Mean-Standard Deviation proportionality.
PPML and GPML both give consistent estimates absentmis-specification ⇒ should be similar if the sample is large enough.
FOC of PPML involves level deviations of Xni from its expectedvalue when OLS involves log deviations. Gamma PML involvespercent deviations ⇒ PPML puts more emphasis on large expectedtrade obs.
Pseudo ML
Xni = exp(zniζ)ηni , where η is a multiplicative error term.Moment conditions of 3 estimators:
zni · (Xni − Xni ) = 0
Poisson
,
zni · (lnXni − ln Xni ) = 0
OLS
,
zni · (Xni/Xni − 1) = 0
Gamma
.
Poisson PML consistent regardless of the distribution of ηni so long asE[Xni | zni ] = exp(zniζ).
Poisson PML is efficient in its class under what we will refer to as the
Poisson Variance Assumption (PVA). var[Xni | zni ] = kE[Xni | zni ],
Data generating processes, Monte #2
Let ui denote a standard normal pseudo-random term.
(a) Lognormal η with low variance parameter (σ = 1):
Xi = exp[−1 lnDisti + 0.5RTAi + ui ]
(b) Lognormal η with high variance parameter (σ = 2):
Xi = exp[−1 lnDisti + 0.5RTAi + 2× ui ]
(c) Mean-variance proportionality (PVA):
Xi = exp[−1 lnDisti + 0.5RTAi + σni × ui ]
σni such that Var[Xi ] = 4× exp[−1 lnDisti + 0.5RTAi ].
(d) Increasing absolute elasticity of distance:
Xni = exp[θi ln Disti + 0.5RTAi + ui ]
θi = −0.5 for Disti < Dist
θi = −1.5 for Disti ≥ Dist.
PML Monte: (a) Log-normal low var
-1.2
-1.1
-1-.
9-.
8M
ean (
1000 r
eps)
of dis
tance c
oef
10 100 1000 10000 100000Sample Size
OLS Gamma PML
Poisson PML
PML Monte: (b) Log-normal high var
-1.2
-1.1
-1-.
9-.
8M
ean (
1000 r
eps)
of dis
tance c
oef
10 100 1000 10000 100000Sample Size
OLS Gamma PML
Poisson PML
PML Monte: (c) Heteroskedastic PV
-1.4
-1.3
-1.2
-1.1
-1M
ean (
1000 r
eps)
of dis
tance c
oef
10 100 1000 10000 100000Sample Size
OLS Gamma PML
Poisson PML
PML Monte: (d) Model mis-specification
-1.1
-1-.
9-.
8-.
7-.
6M
ean (
1000 r
eps)
of dis
tance c
oef
10 100 1000 10000 100000Sample Size
OLS Gamma PML
Poisson PML
Gravity’s errors: cookbook
We recommend all 3 models should be estimated.
1. If 3 estimates are similar, model seems well-specified, error lognormal with a constant variance: Relax.
2. If PPML and GPML are similar and distinct from OLS,heteroskedasticity is probably a problem: OLS unreliable.
3. If Gamma and OLS are similar and Poisson coeffs are smaller inabsolute magnitude (a case we have seen in practice):
(a) RMSE large + small sample: possible small sample bias ofeither the Poisson PML or the Gamma PML.
(b) Large sample: trade costs may have a non-constant elasticity.Major divergence between Poisson and Gamma PML in largesamples is a sign of model mis-specification.
Causes and consequences of zeros
Multiplicative models, in which expected bilateral trade isgiven by E[Xni | zni ] = exp(zniζ), do not naturally generatezero flows.
Actual trade can exhibit substantial fractions of zeros,
Zeros become more frequent with disaggregation at the firmor product level.
Even at the country level, not all small countries tradepositive amounts with other small countries.
So where do the zeros come from?
1. Data reporting issues:
Rounding of small numbers to zero, Declaration thresholds (EU)
2. Demand curves with choke prices Melitz & Ottaviano, ACDR.
3. Bounded productivity combined with fixed costs of market entry asin Helpman, Melitz, & Rubinstein.
4. Sparseness: a finite number of
buyers with heterogeneous preferences shipments (Armenter & Koren’s “Balls & Bins”) industries in the Eaton & Kortum model, exporters in the heterogeneous firms model (with fixed costs)
as in EKS.
And what should do about them? Monte # 3
Monte Carlo estimates of distance and RTA effects for a Xni < σfncensored model.
Estimates: distance RTA Bias (%)Error: Lognormal Poisson Lognormal Poisson Best Worse
LSDV on ln(X ) positives -0.81 -1.07 0.63 0.69 45.19 46.07[0.02] [0.01] [0.06] [0.03]
ET Tobit (ln[a+ X ]) -0.94 -1.06 0.53 0.68 12.41 42.76[0.02] [0.01] [0.06] [0.03]
EK Tobit (ln[Xminn ]) for 0s -0.99 -1.23 0.50 0.57 1.48 35.70
[0.02] [0.01] [0.06] [0.03]PPML (Poisson) -0.73 -1.00 0.29 0.50 0.05 69.65
[0.14] [0.00] [0.43] [0.01]MNPML (Multinomial PML) -0.79 -1.00 0.36 0.50 0.48 49.09
[0.06] [0.02] [0.15] [0.03]Notes: Top value in each cell is the mean estimate (based on 1000 repetitions). The true parameters are -1 fordistance and .5 for RTA. Standard deviation of estimate in “[]”. All estimators include exporter and importerfixed effects.
Comments about Monte # 3
1. “Bad COP” (cond.-on-pos.): ln(X ) omits the zeros.
2. ln(1 + X ) incorporates them but results depend on units of X .Really bad.
3. Tobit-like approach
(a) Eaton & Tamura’s 1994 ln(a+ X ) method. Dominated by....(b) Eaton & Kortum’s 2001 ln(Xmin
n ) replacement. Easy, soundtheory underpinnings, robust.
4. Pseudo ML
(a) Poisson PML: No good under log-normal with high variance.(b) MNPML: Better and theory grounded.(c) Negative Binomial PML: Beware the siren song—it is
unit-dependent too.
Cookbook on zeroes
1. We need to know about the error term distribution.
2. Manning Mullahy test for log-normality mentioned in SST: regresslog of error squared on log predicted trade.
3. Coeff λ should be 2 for log-normal error, 1 for PVA.
4. If λ not sig < 2, EK Tobit is preferred.
5. If λ 2, need to consider PPML or MNPML, with latter beingpreferred (works equally well under poisson errors, less biased andlower variance under log-normal).
Extensive and intensive margins
Usual decomposition (has been refined for MP firms):
∂ lnXni
∂ ln τni=
∂ lnNni
∂ ln τni+
∂ ln xni∂ ln τni
.
Traditional use of “extensive margin”, but the use of“intensive margin” is unconventional.
Mixes what happens to the individual firm with acompositional effect, due to the fact that entrants/exitors aredifferent.
Gravity decomposition (type 1) for France
(1) (2) (3)ln tot exp ln # exp ln av exp
ln GDP (dest) 0.92a 0.58a 0.34a
(0.04) (0.03) (0.02)ln distance -0.95a -0.79a -0.16b
(0.12) (0.10) (0.07)Observations 179 179 179R2 0.818 0.740 0.629
3 Margins
Extensive, intensive, & compositional margins:
∂ lnXni
∂ ln τni=
∂ lnNni
∂ ln τni ext. margin
+1
xni
αni
0
∂ ln xni (α)
∂ ln τnixni (α)
g(α)
G (αni )dα
int. margin
+∂ lnG (αni )
∂ ln αni
∂ ln αni
∂ ln τni
xni (αni )
xni− 1
compos. margin
.
Advantages of 3 margins
Nests existing 2-way decompositions: BJRS/MO (type 1) andChaney (type 2)
Intensive and extensive are “empirically measurable”1. Extensive is elasticity of # of exporters2. Intensive is elasticity of average exports of the “constant
sample” of firms.3. Compositional is a residual
The Pareto-CES margins
Up to now, no functional form assumption. With Pareto andCES:
∂ lnXni
∂ ln τni= −θ
ext. margin
+ 1− σ int. margin
+ σ − 1 compos. margin
.
With 2-way decomposition a la BJRS/MO, int. and comp.margins are added, and ext. has to be 100%.
3-way decomposition enables easy estimation of structuralparameters (using firm-level data).
Future directions for gravity research
1. Digging deeper: Why do the things that matter (distance,language, borders, currencies?) matter as much as they do,even now?
2. Restrictive functional form assumptions: how big of aproblem?
3. Dynamics of gravity
Concluding thoughts
Gravity is an interesting example of1. Strong empirical evidence pushing theory to think about
micro-foundations.2. Theoretical progress changing radically the way those
equations are estimated.
We provide an integrated framework to organise the vastnumbers of micro-foundations
We use quantitative methods to provide cookbook-stylerecommendations about which estimation methods to use andhow to interpret results, notably for policy relevant variables.
Our selective survey of topics at the frontier of currentresearch suggests that a great deal of interesting work liesahead.